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Interstitial gas effect on vibrated granular columns
Javier Pastenes, Jean-Christophe Géminard, Francisco Melo
To cite this version:
Javier Pastenes, Jean-Christophe Géminard, Francisco Melo. Interstitial gas effect on vibrated
gran-ular columns. 2014. �hal-00999226�
Javier C. Pastenes †, Jean-Christophe G´eminard ‡, and Francisco Melo†∗
2
†Departamento de F´ısica Universidad de Santiago de Chile, Avenida Ecuador 3493, 9170124 Estaci´on Central, Santiago, Chile.
‡Laboratoire de Physique, Ecole Normale Sup´erieure de Lyon,
Universit´e de Lyon, CNRS, UMR 5672, 46 All´ee d’Italie, F-69007 Lyon, France.
3
(Dated: June 4, 2014)
4
Vibrated granular materials have been intensively used to investigate particle segregation,
convec-5
tion and heaping. We report on the behavior of a column of heavy grains bouncing on an oscillating
6
solid surface. Measurements indicate that, for weak effects of the interstitial gas, the temporal
vari-7
ations of the pressure at the base of the column are satisfactorily described by considering that the
8
column, in spite of the observed dilation, behaves like a porous solid. In addition, direct observation
9
of the column dynamics shows that the grains of the upper and lower surfaces are in free fall in the
10
gravitational field and that the dilation is due to a small delay between their takeoff times.
11
PACS numbers: 45.70.Mg, 45.70.Qj, 81.20.Ev.
12
I. INTRODUCTION
13
The rapid compression of a relatively loose pile of sand
14
or of snow may require a high pressure to drive the
15
flow of the interstitial fluid between the solid particles,
16
grains or flakes. The effect, together with the elastic
17
and frictional resistance, contributes to the pressure to
18
overcome to compress the material. Interestingly, due
19
to this drainage effect, snowboarding and sandboarding
20
benefit from a significant lift force and therefore from a
21
significant reduction of the friction at large slip velocity
22
if the medium is loose enough [1]. Indeed, viscous forces
23
are prone to be at play when a gas is evacuated through
24
a wide variety of porous materials frequently found in
25
common life and industrial applications [2]. From
phys-26
ical viewpoint the influence of interstitial viscous forces
27
on non-cohesive granular materials has generated
long-28
lasting debate due mainly to the difficulties introduced
29
by the complex rheology of unconsolidated porous media,
30
and by the sensibility of the response to the conditions
31
imposed at the boundary surfaces. Booming sand [3, 4]
32
and the jets resulting from the impact of a solid object
33
onto the surface of a loosely packed granular bed [5–7]
34
are subtle manifestations of the coupling of the
mechan-35
ical response of granular matter with the dynamics of
36
the interstitial fluid. Heaping, granular convection and
37
size segregation under vibration [8–10] are a few other
38
examples of phenomena in which the internal viscous
39
forces drive, at least partially, the motion of the grains
40
and, thus likely, changes in the external shape of the
sys-41
tem [11, 12].
42
In the same way, in thin layers of non-cohesive powders
43
submitted to repeated pats, granular droplets appear as
44
a result of the interplay between the air flow through the
45
material, which leads the droplets to grow, and the
sta-46
bility of the granular slopes, which limits their size [13].
47
∗francisco.melo@usach.cl
In a previous work, we reported on the formation and
48
on the, even more striking, upward motion of millimetric
49
droplets on an incline subjected to vertical vibration [14].
50
We later showed that the viscous drag, which is of the
51
order of the droplet weight, is responsible for the droplet
52
formation while the gas pressure at the droplet base
pro-53
vides an effective horizontal acceleration whose
cumula-54
tive effect is an upward displacement of the center of mass
55
after each cycle of the vibration [15]. Interestingly, the
56
experiments revealed that the droplets move only if the
57
maximum acceleration of the substrate is larger than a
58
threshold which we associated, in a first qualitative
ap-59
proach, to a characteristic dilation.
60
In the present report, we focus on the gas pressure and
61
dilation in a simplified geometry, i.e. a cylindrical
gran-62
ular column subjected to vertical vibration. We limit the
63
study to the regime of low viscous friction by using
par-64
ticles of relatively large size and low frequency of
vibra-65
tion. The main aim of the study is to provide insight into
66
the mechanisms that lead the column to dilate. First, we
67
show that a classical Darcy’s law accounts for the
dynam-68
ics of the gas pressure at a column base. Interestingly, the
69
agreement of our measurements with early predictions
70
obtained by assuming a rigid porous medium [16, 17],
71
indicates that, for sufficiently tall columns, the porosity
72
changes associated with the column dilation have
neg-73
ligible effects. However, even in this limit, a significant
74
overall dilation of the column is observed. From the
addi-75
tional detailed analysis of the system dynamics, we
con-76
clude that the granular column not only does not dilate
77
along its whole height but also that, indeed, the
dila-78
tion only involves the grains of the lower and upper
sur-79
faces, which experience slightly delayed free falls. Our
80
results provide a more quantitative way to assess the
81
dilation effects and the role they play in the
instabili-82
ties observed in related systems, such as those mentioned
83
hereinabove.
2
FIG. 1. (Color online) Sketch of the experimental device – The grains inside lye inside a cylindrical container vibrated vertically. The resulting pressure variations in the gap be-tween the substrate and the bottom surface of the column, ∆P , is monitored by means of a differential pressure trans-ducer (DPT) while a high-speed camera is used to observe the dynamics of the column from the side. Bottom-right in-set: Details of the L-shaped tube connecting the gap to the DPT and of the grid at the surface of the mount. Top-right inset: Typical images from the camera (a) Initial contact be-tween the column and the substrate, previous to take-off (b) Large gap underneath the column in flight (c) Sudden land-ing of the column [Steel grains, d = 745 µm, h0 = 5.7 mm,
f = 15 Hz and Γ = 2.6].
II. EXPERIMENTAL SETUP AND PROTOCOL
85
The experiment consists in monitoring the dynamics
86
and the pressure at the base of a granular material placed
87
inside a vertically vibrated cylindrical vessel.
88
The container is made of a transparent Plexiglass tube
89
(Height: 46 mm; Inner radius: 10 mm), glued to a rigid
90
metallic mount (aluminum alloy) as sketched in Fig. 1.
91
It is filled with steel beads [diameter d = (465 ± 73) µm
92
and density ρs= (7.4 ± 0.2) 10
3 kg·m−3] up to an initial
93
height, h0, ranging from 2.5 mm and 18 mm at rest. The
94
inner diameter of the container is more than 20 times the
95
grain diameter, which insures that the finite-size effects
96
due to the lateral wall are negligible. The lid at the top
97
leaves the air enter freely in the tube. An internal
L-98
shaped pipe, drilled in the mount (radius rp = 1 mm),
99
makes it possible to measure the pressure of the gas
un-100
derneath the column. At one end, a grid (45 µm,
usu-101
ally used for Transmission Electron Microscopy) avoids
102
that the grains enter inside the tube while insuring the
103
continuity of the gas pressure. At the other end, the
104
tube is connected to a differential pressure transducer
105
(DPT, Omega, PX277) through a non-torsional hose,
106
which avoids pressure variations due to the deformations.
107
We checked that the response time of the transducer is
108
shorter than 1 ms. Thus, the configuration achieves
mea-109
surement of the pressure difference, ∆P , with an
accu-110
racy of about 2 Pa in the range ±124 Pa.
111
The whole is vibrated vertically using an
electrody-112
namic exciter (Labworks, MT-160) fed with a sinusoidal
113
current of frequency, f , in the range 15 to 50 Hz. The
114
acceleration of the container, γ(t), is monitored by means
115
of a charge accelerometer, placed at the top, its axis
116
aligned with the vertical. From the signal, γ(t), we
de-117
termine, to within 0.01, the dimensionless acceleration
118
Γ ≡ max (γ)/g = Aω2/g, where A stands for the
am-119
plitude of the vibration and g for the magnitude of the
120
acceleration due to gravity (ω ≡ 2π f ). In the present
121
study, Γ is chosen within the range from 1 to 4.
122
The dynamics of the granular material is observed from
123
the side by means of High Speed (HS) video camera.
124
The resolution of the images is of 256×256 px2
together
125
with an acquisition rate of 1200 fps. The heights, z0and
126
z1, of the free surface and of the bottom of the column,
127
respectively, are obtained with a resolution of 0.2 mm by
128
elementary image analysis.
129
III. EXPERIMENTAL RESULTS
130
A. General description
131
For given vibration frequency f and dimensionless
ac-132
celeration Γ, we report on the dynamics of the granular
133
column and on the temporal evolution of the pressure
134
∆P in the steady state (Fig. 2).
135 136
First, the dynamics of the column is mainly
charac-137
terized by the vertical positions, z0(t) and z1(t), of its
138
upper and lower surfaces (Fig. 2a). One observes that,
139
on the one hand, the column periodically looses contact
140
with the substrate, which is better illustrated by
display-141
ing the gap, s(t) ≡ z1(t) − z(t), i.e. the vertical size of
142
the region free of grains between the substrate and the
143
column (Fig. 2b). On the other hand, the column
period-144
ically dilates, which is clearly revealed by reporting the
145
column height, h(t) ≡ z0(t) − z1(t) (Fig. 2c). The signal
146
from the accelerometer exhibits a significant noise after
147
the gap has vanished until the dilated column recovers
148
its initial height (Fig. 2d) A complex temporal evolution
149
of the pressure ∆P (t) results from the dynamics of the
150
grains (Fig. 2e).
151
In next section III B, we interpret qualitatively the
152
behavior of the system. In section III C, we discuss
153
thoroughly the temporal behavior of the pressure signal,
154
∆P (t), whereas section III D is devoted to the dynamics
155
of the granular column.
156
B. Qualitative understanding
157
Let us first assume that the column sits at rest on the
158
substrate and that the pressure inside is in equilibrium
FIG. 2. (Color online) Evolution of column characteristics and of the pressure as a function of phase ωt – (a) Vertical positions of the substrate z (continuous line), of the upper sur-face z0(full squares) and of the lower surface z1(open squares)
vs. phase ωt. Dashed-dotted line: h0+ z is a guide for the
eye. Red (light gray) thick line and Blue (dark gray) dashed line: free fall of the upper and lower surface respectively. The parabolas have curvature -g. (b) Gap s(t) ≡ z1(t) − z(t) –
In region I (blue), the column is not in contact with the sub-strate. (c) Column height h(t) ≡ z0(t) − z1(t) – The column
exhibits a significant dilation in regions I (blue) and II (yel-low). Straight line: linear increase of h. (d) Acceleration γ(t) – The significant noisy vibration in region II (yellow) is due to the collapse of the column onto the substrate. Red (light gray) circle: γ = −1. (e) Pressure ∆P – In region I (blue), while the column takes off and dilates, ∆P decreases, reaches a minimum and increases again. In region II (yellow), ∆P continues to increase while the column, in contact with the substrate, settles back. In a last phase, in region III (red), ∆P decreases while the column seats at rest on the substrate. [h0= 5.7 mm, f = 15 Hz and Γ = 1.81].
with the outer pressure. Provided that the typical
veloc-160
ity associated with the vibration Aω is smaller than the
161
speed of sound in air, the vibration does not induce any
162
significant variation of the pressure, ∆P , if the grains do
163
not move. This stage lasts as long as the weight of the
col-164
umn insures the contact with the substrate, i.e. as long
165
as the downward acceleration of the substrate does not
166
exceed the acceleration due to gravity. In other words,
167
nothing happens as long as −γ(t) < g or, equivalently,
168
γ/g > −1.
169
1. Take-off and flight
170
When γ/g . −1, the acceleration due to gravity does
171
not insure the contact anymore and the column starts to
172
take off. The system enters region I in Fig. 2. However,
173
the column, as a whole, does not experience a free flight.
174
Indeed, the take-off requires the opening of a gap between
175
the column and the substrate, which corresponds to an
176
increase of the volume of the gas in the gap region and,
177
thus, induces a decrease of the local pressure (Fig. 2e,
178
region I). In turn, the column is subjected to a pressure
179
force which partially impedes the opening of the gap.
180
However, it is interesting to notice that, provided that
181
the viscous drag on individual grains is negligible [18],
182
the grains of the free surface are almost free to move and
183
to take off at γ/g = −1 whereas, by contrast, the grains
184
at the bottom are constrained by the column above. As
185
a consequence, the column starts to dilate (Fig. 2c,
re-186
gion I)
187
In order to understand why the pressure ∆P exhibits a
188
minimum during the column flight above the substrate,
189
one must remark that the pressure difference between the
190
upper and bottom surfaces induces a gas flow through the
191
column which is indeed permeable. The pressure
evolu-192
tion is thus the result of the competition between the
193
volume expansion, due to the opening of the gap, which
194
leads to a decrease of ∆P and the inflow, due to the
per-195
meability of the column, which leads to a relaxation of
196
∆P toward the equilibrium with the outside pressure. In
197
our experimental conditions, the observation of a
min-198
imum in ∆P reveals that the characteristic relaxation
199
time, τris of the order of the flight duration (itself of the
200
order of 1/f in the reported example).
201
2. Sudden landing
202
Due to its fall in the gravity field and to the vertical
vi-203
bration of the container, the lower surface of the column
204
enters again in contact with the substrate. The system
205
enters region II in Fig. 2. The height h(t) of the column
206
then rapidly recovers its initial value h0 (Fig. 2c). This
207
collapse of the column produces the noise seen in the
208
signal from the accelerometer (Fig. 2d). Provided that
209
the pressure relaxation time, τr, associated with the gas
210
transport in the column, is larger than the typical
col-211
lision time, τc, the pressure, ∆P , still increases as long
212
as the height of the column decreases (Fig. 2d). As a
4
consequence, the maximum of ∆P is not reached at the
214
collision time but later on, close to the end of the column
215
collapse.
216
3. Relaxation
217
Finally, after the collapse, the column sits at rest on
218
the substrate. The system enters the region III in Fig. 2.
219
However, the pressure of the gas in the column is initially
220
larger than the outer pressure. It relaxes continuously,
221
with a characteristic time τr, toward the outside
pres-222
sure because of the resulting gas flow through the grains
223
(Fig. 2e) until the next take-off (Sec. III B 1).
224
C. Pressure pattern, ∆P (t)
225
Here, we introduce a theoretical framework to support
226
the qualitative description proposed in Sec. III B.
227
1. Take-off and flight
228
In a first simplified approach, we consider that the
col-229
umn moves as a whole and we neglect the dilation and
230
the possible grain convection. If the inner pressure is
ini-231
tially in equilibrium with the outer pressure, the column
232
takes off when the downward acceleration of the substrate
233
equals that of the gravity, thus for γ = −g. The column
234
is subsequently flying.
235
In flight, the column is submitted the gravity and to the
236
force associated with ∆P . Denoting zG(t) the altitude of
237
the column center of mass, G, we write:
238 d2 zG dt2 = −g + 1 ρh0 ∆P (t). (1) 239
This equation explicitly couples the dynamics of the
col-240
umn with the overpressure ∆P . However, note that the
241
gas pressure alters the dynamics only if ∆P is of the
or-242
der of ρgh0, the stress applied by the column onto the
243
substrate at rest.
244
Now, in order to account for the pressure variations
in-245
duced by the column dynamics, we consider that ∆P
246
induces a gas flow through the grains. The
instan-247
taneous flow-rate is approximately given by a Darcy
248
law, q = −(κ/η)∇P , where η is the gas viscosity and
249
κ the permeability given by the Ergun relation, κ =
250
ψ3
d2
/[150(1 − ψ)2
], where ψ is the porosity [18].
As-251
suming further that the gas is incompressible, we
esti-252
mate that the variation of the gap s(t) between the
col-253
umn and the substrate is only permitted by the gas flow,
254
which imposes that ds/dt = q, with q = (κ/η)(∆P /h0)
255
from the Darcy law applied to our configuration. We
256 thus have: 257 d∆P dt = h0 η κ ds dt. (2) 258
Thus, combining the equations governing the motion of
259
the column (Eq. 1) and the pressure variations (Eq. 2)
260
and taking into account that, in absence of dilation, zG =
261 h0/2 + s + z, we write: 262 d2 ˜ s dφ2 + 1 ˜ φκ d˜s dφ = sin(φ + φ0) − 1 Γ, (3) 263
where ˜s ≡ s/A, φ ≡ ωt and φ0≡ωt0= arcsin 1/Γ, t0
be-264
ing the time of the take-off [i.e. γ(t0) = −g]. The
param-265
eter ˜φκ≡ωκρ/η is a relaxation time expressed in units of
266
the vibration period. Eq. (3) was first obtained by Kroll
267
for a porous oscillating piston in his pioneering works [16]
268
and it is referred to as the Kroll’s model. Eq. (3) has an
269
analytic solution which is written [10]:
270 ∆P (φ) = − ρgh0 1 + ˜φ2 κ hp Γ2−1 (sin φ − ˜φ κcos φ) + ˜φκsin φ − ˜φ 2 κ+ cos φ + ˜φκ( p Γ2−1 + ˜φ κ) e −˜φ φκ −1i. (4)
The relaxation time ˜φκis the characteristic time needed
271
by the column to reach the regime governed by the air
272
viscosity. For small fluid viscosity η, large density ρ of the
273
material the grains are made of, or large grain diameter
274
d (the porosity scales like d2), the effect of air is tiny and
275
this time can be large in comparison with the period of
276
the vibration. In the limit ˜φκ≫1, the pressure difference
277
∆P (φ) in Eq. (4) exhibits the minimum:
278 ∆Pmin ρgh0 = − 1 ˜ φκ " arccos 2 Γ2 −1 −2pΓ2−1 # (5) 279
Interestingly, ∆Pmin depends on one single adjustable
280
parameter, ˜φκ, provided that the acceleration Γ and
281
the weight ρgh0 (per unit area) of the column are
282
known.
283
In Fig. 3a, we report ∆Pmin/(ρsgh0) as a function of
284
Γ for various column height h0 (As the porosity ψ and,
285
thus the density of the column ρ = (1 − ψ) ρs, are a
pri-286
ori unknown, we normalized the data using the density
287
of steel ρs). First, we observe an excellent collapse of the
288
data on a master curve, except for the thinnest column
289
at large acceleration (h0 = 2.1 mm and Γ > 2.5). When
290
the column is too thin and the acceleration too large, the
291
grains do not bounce as a whole but rather form a gaseous
292
phase and, then, the model fails in describing the
pres-293
sure pattern, ∆P (t). Except for the thinnest column, the
294
interpolation of the experimental data with Eq. (5) leads
295
to ˜φκ= (14.6 ± 0.1) and, thus, to ψ ≃ 0.51 (we consider
296
the viscosity of air η = 18.6 10−6 Pa s). The porosity is
297
found to be greater than the porosity of a random loose
298
packing, which is acceptable for a column flying almost
299
freely, not compacted by gravity. The dependence on
fre-300
quency of ∆Pmin at constant Γ constitutes an additional
301
clue that the model is acceptable (Fig. 3b). Note finally
FIG. 3. (Color online) Normalized minimum gap-pressure, ∆Pmin/(ρsgh0): (a) dependence on acceleration Γ at constant
frequency f = 15 Hz. (b) dependence on frequency at con-stant acceleration Γ = 2.16 Hz. Solid line: fit from Eq. (5) with ˜φκ= 14.6 ± 0.1, which leads to ψ ≃ 0.51.
that the model remains valid even if the characteristic
303
(normalized) time ˜φκ is not much larger than the unity.
304
Nevertheless, the rather large value of ˜φκ indicates that
305
the viscosity almost does not alter the trajectory of the
306
column that should nearly experience a free flight. The
307
assumption will be discussed in Sec. III D.
308
2. Layer at rest
309
After the column-substrate collision, the column
col-310
lapses and then sits at rest on the solid surface, the
in-311
ner pressure being initially larger than the outer pressure
312
(Fig. 2e, left of region III). We observe that ∆P slowly
313
relaxes towards 0. However, our crude model cannot
ac-314
count for this relaxation as ∆P is expected to vanish
315
when the column moves with the substrate (Darcy law,
316
Sec. III C 1). We previously assumed that the
compress-317
ibility of the gas could be neglected when the grains are
318
in motion (Sec. III C 1), but we must take it into account
319
to describe the relaxation of ∆P when the column is at
320
rest.
321
Considering the Darcy law and the adiabatic pressure
322
variation due to the associated gas flow in a granular
323
column of porosity ψ, we write the diffusion coefficient
324
D = αP0κ/[η(1 − ψ)], where P0 stands for the outside
325
pressure and α = 1.4 for the adiabatic constant for dry
326
air. The typical relaxation time in a column of height
327
h0 is τ = h20/D. In our experimental conditions, taking
328
ψ = 0.58 for the column sitting at rest on the substrate,
329
we estimate D ≃ 3 m2
/s. For h0 = 5.7 mm, we thus
330
get τ ∼ 10 µs, much shorter than the time observed
331
experimentally.
332
In order to recover the experimental relaxation time,
333
one must take into account that the column sits above a
334
pressurized cavity and that the relaxation time is rather
335
due to the escape of the gas trapped underneath. We
es-336
timate that the total volume of the L-shaped pipe drilled
337
in the tube mount and of the hose connecting the latter
338
to the pressure transducer, vconn. ∼ 2 cm
3. Assuming
339
that the gas escapes only through a cylinder of length h0
340
and radius rp within the column, we expect the
result-341
ing characteristic time τ = ηh0vconn./(πr 2
pαP0κ) to be
342
about 30 ms for h0 = 5.7 mm. This estimate is of the
343
order of the typical relaxation time, of about 5 ms, which
344
is observed experimentally (Fig. 2a). Assuming that the
345
gas escapes only through a tube of radius rp obviously
346
leads to an overestimate but the agreement validates the
347
proposed mechanism of relaxation.
348
3. Discussion of the pressure pattern
349
We have seen that the pressure pattern is reasonably
350
described by considering two different regimes. In
re-351
gion I, after take-off, the decrease of the pressure, ∆P ,
352
and its minimum are recovered by using a Darcy law,
353
while neglecting the compressibility of the gas and the
354
dilation of the column. In region III, the relaxation of
355
the pressure requires the compressibility of the gas to be
356
considered.
357
In this framework, the evolution of ∆P while the
col-358
umn settles back onto the substrate (Fig. 2, region II)
359
would require to take both the dilation of the column
360
and the compressibility of the gas into consideration.
361
We mention here that, in this regime, a horizontal front
362
separates a column of grains sitting at rest on the
sub-363
strate from the grains above that are still in motion. The
364
description proposed in Sec. III C 1 should remain valid
365
when applied to the grains in motion. This argument
366
at least explains the continuity of the pressure evolution
367
when the column hits the substrate. Indeed, there is no
368
discontinuity of the velocity at the beginning of the
con-369
tact. In addition, after the contact, the height of the
370
column of grains that are still in motion decreases which
371
explains that the contribution of the grain motion to the
372
pressure variation d∆P/dt (Eq. 2) decreases. At the same
373
time, the pressure relaxes towards the outer pressure as
374
explained previously in Sec. III C 2. As a result of the
375
two effects, the pressure reaches a maximum somewhere
376
in the region II (Fig. 2), before the column completely
377
collapsed and remains sitting at rest on the substrate.
378
At this stage we compare the pressure pattern to
for-379
mer works by Gutman [17]. Indeed, Gutman extended
6
the simplified Kroll’s model to account for the gas
com-381
pressibility upon the gas flow through a porous layer and
382
performed pressure measurements beneath the vibrated
383
layer. Although Gutman did not consider the
possibil-384
ity of layer dilation on his model, the calculated pattern
385
contains the main features we observed experimentally
386
(compare Fig. 2 to Fig. 2 in Ref. [17]). The main feature
387
attributed to compressibility effects is that the decay of
388
the air pressure in the column after the collision takes a
389
finite time, so that when the column takes off in the next
390
cycle the gas pressure in the opening gap is above
atmo-391
spheric. The effect is not significant in our experimental
392
conditions [19].
393
Finally, we point out that the measurements of ∆P
394
during the take-off, and direct measurements of the
sub-395
sequent flight time, indicate that the trajectory of the
396
column is not different from that of a porous solid (for
397
Γ < 3)[20, 21]. One can thus wonder how it is then
possi-398
ble to understand that this result is compatible with the
399
observation of a significant dilation. The question will be
400
answered in the next section, in which we even propose
401
a dilation mechanism.
402
D. Layer Dilation
403
In Fig. 2c, one observes that the column dilates
dur-404
ing its flight (region I). The dilation of the column can
405
be accounted for, by considering that the behavior of
406
the grains at the upper and lower surfaces differs
qual-407
itatively from that of the grains in the bulk of the
col-408
umn. Indeed, at the surface, the grains, in addition to
409
the mechanical solid contact with their neighbors below
410
and above, are submitted to gravity and to the friction
411
with air which is small and, negligible in our
experimen-412
tal conditions.
413
Consider the grains of the first layer at the top of the
414
column. We observe experimentally that they experience
415
a free fall, z0(t) (Fig. 2a). To account for this observation,
416
we note that the friction of air has negligible effect on
iso-417
lated grains or, at least, an effect much smaller than that
418
on a dense column. As a result, at γ = −1, the grains
419
of the free surface take off and detach from the dense
420
column below whose trajectory, governed by Eq. (4), is
421
always below that expected for a free fall. As a
conse-422
quence, z0= A sin (ωt0) + A ω cos (ωt0) (t − t0) − 1 2g (t −
423
t0)2 where, we remind, t0 is the time at take-off.
424
Interestingly, we observe in Fig. 2c that h increases
lin-425
early with time t. The height h being defined as the
426
difference between the altitude z0 of the upper and z1
427
lower surfaces, we conclude that the grains at the
bot-428
tom also experience a parabolic flight with the same
429
acceleration, thus a free fall. This conclusion is
sup-430
ported by the direct observation of the free fall in Fig. 2a,
431
where both (upper and lower) parabolas have curvature
432
-g. The observed linear increase of h with time thus
re-433
sults from the fact that the free falls of the grains at
434
the upper and lower surfaces do not have the same
ini-435
FIG. 4. Trajectory of the column bottom layer: Dimension-less free fall motion model, z1/A, for different time delays
(dotted line: δt = 1 ms, small N : δt = 3 ms, dashed line: δt = 5 ms) and Eq. (1) trajectory estimation, s+z (solid black line). Open crossed squares: z/A. [Γ = 1.81 and f = 15 Hz].
tial conditions. Taking t1 as the origin of the free fall of
436
the lower layer we can assume that the initial position
437
and velocity are those of the substrate at time t1, i.e.
438
z1 = A sin (ωt1) + A ω cos (ωt1) (t − t1) − 1
2g (t − t1)2.
439
Doing so, we expect a linear increase of h with the
veloc-440 ity: 441 dh dt = 1 2A r 1 − 1 Γ2ω 2 δt2 (6) 442
where we define δt = t1−t0, the delay between the origins
443
of the free falls of the lower and upper surfaces. From
444
the experimental slope, we get δt = (4.7 ± 0.2) ms.
445
It is then particularly interesting to discuss the physical
446
origin of the delay. We already observed that the grains of
447
the lower surface experience a free fall. One must however
448
notice that the grains can be in free fall only if their
449
motion is not frustrated. Note that, when they take off,
450
their position and velocity are limited by the solid surface
451
below and the grains above. Their velocity is oriented
452
upwards and their acceleration equals the acceleration
453
due to gravity only if their trajectory does not intersect
454
the trajectory of the grains above. In Fig. 4, we report
455
the trajectory of the bottom layer, z1(t), in free fall for
456
several values of δt (taking t1= t0+δt), and the altitude,
457
z(t) + s(t), estimated from the solution of Eq. (1) (black
458
line in Fig. 4). We observe that for small δt, z1> z + s,
459
which means that the motion of the grains of the bottom
460
surface is limited by the motion of the grains above (δt =
461
1 ms, dotted line in Fig. 4). On the contrary, for large
462
enough δt, z1< z + s at all time until the collision with
463
the substrate. The grains can experience a free fall (δt =
464
5 ms, dashed line in Fig. 4). For intermediate values of
465
δt, the trajectories, z1 and s + z, cross each other at a
466
time which compares with the collision time (δt = 3 ms,
467
small triangles in Fig. 4). The grains can experience a
468
trajectory very similar to a free fall until it collides with
substrate. From the latter simple observation, one can
470
deduce that a delay of, at least, 3 ms is necessary for the
471
grains of the lower surface to fall freely and that 5 ms is
472
clearly an overestimate of δt.
473
Thus, the simple argument above gives a reasonable
474
range, 3 to 5 ms, for the experimental delay δt = 4.7 ms,
475
which validates the potential mechanism proposed to
ac-476
count for the dilation. In summary, the grains of the
477
two free surfaces of the column experience free falls, the
478
take-off of the lower grains being delayed by the presence
479
of the dense column above which experience a
trajec-480
tory governed by the interplay between the acceleration
481
of gravity and the friction with the gaseous phase.
482
E. Conclusion
483
In conclusion, we observed the bouncing of a porous
484
column of grains and measured the resulting variation
485
of the pressure underneath. When interaction between
486
the column and the gas are weak, because of the size
487
and weight of the grains, the pressure is reasonably
ac-488
counted for by considering the column as a porous solid,
489
thus neglecting the column dilation. The latter is
satis-490
factorily explained by considering that the grains of the
491
upper and lower surfaces experience a free falls. In this
492
framework, the dilation only results from a delay between
493
the departure times and not from any pressure profile
494
within the column that would repel the grains from one
495
another.
496
ACKNOWLEDGMENTS
497
The authors acknowledge the financial support from
498
the contracts ANR-09-BLAN-0389-01/Conicyt-011 and
499
CNRS-Conicyt-PCCI12016.
500
[1] Q. Wu, Y. Andreopoulos, and S. Weinbaum, Physical
501
Review Letters 93, 19, 194501, (2004).
502
[2] See for instance, F. J. Muzzio, T. Shinbrot and B. J.
503
Glasser, Powder Technology 124, 1, (2002).
504
[3] R. A. Bagnold, The Physics of Blown Sand and Desert
505
Dunes. (Methuen, London, 1954).
506
[4] B. Andreotti, L. Bonneau, and E. Cl´ement, Geophys.
507
Res. Lett., 35, L08306, (2008).
508
[5] S. T. Thoroddsen and A. Q. Shen, Phys. Fluids, 13, 4,
509
(2001).
510
[6] D. Lohse, R. Rauhe, R. Bergmann and D. Van Der Meer,
511
Nature, 432, 689, (2004).
512
[7] S. Deboeuf, P. Gondret and M. Rabaud, Phys. Rev. E,
513
79, 041306, (2009).
514
[8] C. Laroche, S. Douady and S. Fauve, J. Phys. (Paris),
515
50, 699, (1989).
516
[9] P. Evesque and J. Rajchenbach, Physical Review Letters
517
62, 44, (1989).
518
[10] L.I. Reyes, I. S´anchez, G. Guti´errez, Physica A. 358, 466,
519
(2005).
520
[11] H. K. Pak and R. P. Behringer, Physical Review Letters
521
71, 1832, (1993).
522
[12] H. K. Pak, E. Van Doorn, and R. P. Behringer, Physical
523
Review Letters 74, 4643, (1995).
524
[13] J. Duran, Physical Review Letters 87, 254301, (2001).
525
[14] L. Caballero and F. Melo, Physical Review Letters 93,
526
258001, (2004).
527
[15] J. C. Pastenes, J.-C. G´eminard and F. Melo, Phys. Rev.
528
E 88, 012201 (2013).
529
[16] W. Kroll, Forsch. auf der Geb. des Ing. 20, 2, (1854).
530
[17] R. G. Gutman, Trans. Instn. Chem. Engrs, 54, 174-183,
531
(1976).
532
[18] See for instance, R.M. Nedderman, Statics and
Kinemat-533
ics of Granular Materials, (Cambridge Univ. Press 1992).
534
[19] Compressibility effect can be neglected if τD/τS << 1,
535
where τS is the time during which the column sits on the
536
vibrating surface and τD= hhi 2
/D, with D ≡ κ/ψηχ the
537
relevant diffusion coefficient and χ the gas
compressibil-538
ity. Note that τS is a decreasing function of both f and
539
Γ. In our experimental conditions, for values of Γ . 3,
540
compressibility effects become significant for Γ & 100 Hz.
541
[20] Notice that this result remains accurate for values of
542
Γ . 3. Above this value, the flight time begins to deviate
543
from the prediction of the inelastic ball model.
Indepen-544
dent work indicates that this effect develops if the sitting
545
time of the column on the vibrating plate becomes of the
546
order of the collision time [21].
547
[21] J.M. Pastor, D. Maza, I. Zuriguel, A. Garcimart´ın and
548
J.-F. Boudet, Physica D, 232, 128135, (2007).
